A kite is flying at the height of 75 m from the ground .The string makes an angle `theta` (where `cottheta=(8)/(15))` with the level ground .As - summing that there is no slack in the string the length of the string is equal to :

Options

A. 85 metre
B. 65 metre
C. 75 metre
D. 40 metre

Answer

To find the length of the string of the kite flying at a height of 75 m, we can use trigonometric relationships and the Pythagorean theorem. Let's break down the solution step by step.

### Step 1: Understand the given information
We know that:
- The height of the kite (perpendicular) = 75 m
- The angle θ is such that cot(θ) = 8/15

### Step 2: Relate cotangent to the triangle
From the definition of cotangent:
\[ \cot(\theta) = \frac{\text{base}}{\text{perpendicular}} = \frac{BC}{AB} \]
Here, \( AB \) is the height (75 m), and \( BC \) is the horizontal distance from the point directly below the kite to the point where the string is held.

### Step 3: Substitute the values
Given that:
\[ \cot(\theta) = \frac{8}{15} \]
We can express this as:
\[ \frac{BC}{75} = \frac{8}{15} \]

### Step 4: Solve for BC
Cross-multiplying gives:
\[ BC = 75 \cdot \frac{8}{15} \]
Calculating this:
\[ BC = 75 \cdot \frac{8}{15} = 75 \cdot 0.5333 = 40 \text{ m} \]

### Step 5: Use the Pythagorean theorem
Now we have:
- \( AB = 75 \) m (height)
- \( BC = 40 \) m (horizontal distance)

We need to find the length of the string (hypotenuse \( AC \)):
\[ AC = \sqrt{AB^2 + BC^2} \]
Substituting the values:
\[ AC = \sqrt{75^2 + 40^2} \]
Calculating \( 75^2 \) and \( 40^2 \):
\[ 75^2 = 5625, \quad 40^2 = 1600 \]
Thus:
\[ AC = \sqrt{5625 + 1600} = \sqrt{7225} \]

### Step 6: Calculate the square root
Finding the square root:
\[ AC = 85 \text{ m} \]

### Conclusion
The length of the string is \( 85 \) meters.

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